Dr.ir. R. Versendaal
Dr.ir. R. Versendaal
I have a passion for probability theory. My interest is both theoretical in nature, as well as aimed towards applications of probability theory in other areas of research. In particular, I combine probability theory with the fields of differential geometry and graph theory.
Stochastic processes in manifolds
One of my main interests is to study the behavior of stochastic processes in (Riemannian) manifolds. Here, one can for instance think about random walks and diffusions such as Brownian motion. My research mainly focuses on their asymptotic behavior, in particular on large deviations. Large deviations are concerned with the limiting behaviour on the exponential scale of a sequence of random variables. Usually, this sequence of random variables satisfies a law of large numbers, and deviations are considered 'large' if it are deviations on this scale. However, I am also interested in asymptotics on different scales, such as central limit type theorems. This work also extends to evolving Riemannian manifolds, in which the Riemannian metric is time-dependent. One can for instance think about inflating a balloon or for instance the evolution of a (biological) cell. A well known example of a time-dependent Riemannian metric is the Ricci-flow, which intuitively homogenizes the curvature of the manifold. However, I do not necessarily focus on specific examples, and the results are valid under some general regularity conditions of the time-dependence.
Random graphs and geometry
Another line of research I am interested in is the behavior of random graphs, in particular with some spatial constraints. So far, I have worked on analyzing such random graph models when also a degree sequence is given. Since this degree constraint competes with the spatial constraint, this raises the necessary complications. Furthermore, linking this to the first topic, such random graphs may serve as discrete approximations of Riemannian manifolds. Finding good such discretizations is typically challenging. However, many stochastic processes rely on having some sort of discrete structure on which they can be defined. This is for instance the case for interacting particle systems, which by taking a suitable discretization, can then be studied also in Riemannian manifolds.
Classical large deviation theorems on complete Riemannian manifolds
Richard C. Kraaij / Frank Redig / Rik Versendaal
Large deviations for geodesic random walks
Lp-Analysis of the Hodge–Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds.
Jan van Neerven / Rik Versendaal
2022-05-01 - 2024-05-01